Theorem ssrabdv 3327

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)

Hypotheses

Ref	Expression
ssrabdv.1	|- ( ph -> B C_ A )
ssrabdv.2	|- ( ( ph /\ x e. B ) -> ps )

Assertion

Ref	Expression
ssrabdv	|- ( ph -> B C_ { x e. A | ps } )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem ssrabdv

Step	Hyp	Ref	Expression
1		ssrabdv.1	. 2 |- ( ph -> B C_ A )
2		ssrabdv.2	. . 3 |- ( ( ph /\ x e. B ) -> ps )
3	2	ralrimiva 2623	. 2 |- ( ph -> A. x e. B ps )
4		ssrab 3326	. 2 |- ( B C_ { x e. A | ps } <-> ( B C_ A /\ A. x e. B ps ) )
5	1, 3, 4	sylanbrc 421	1 |- ( ph -> B C_ { x e. A | ps } )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   A.wral 2528   {crab 2532   C_ wss 3220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-in 3226  df-ss 3233

This theorem is referenced by:  perfectlem2  16031